PRICING EQUITY DERIVATIVES WITH THE KRISTENSEN-MELE ...

PRICING EQUITY DERIVATIVES WITH THE KRISTENSEN-MELE APPROACH

Carlos Catalán García

Trabajo de investigación 013/007

Master en Banca y Finanzas Cuantitativas

Tutores: Dr. Manuel Moreno Dr. Javier F. Navas

Universidad Complutense de Madrid

Universidad del País Vasco

Universidad de Valencia

Universidad de Castilla-La Mancha

www.finanzascuantitativas.com

Pricing equity derivatives with the

Kristensen-Mele approach

Carlos A. Catalán Garcia

June 28, 2013

Advisors:

Manuel Moreno Fuentes (Universidad de Castilla-La Mancha)

Javier Fernández Navas (Universidad Pablo de Olavide de Sevilla)

Master Dissertation

Master in Banking and Quantitative Finance

1

Agradecimientos.

Me gustaría agradecer a mis tutores, Manuel Moreno Fuentes y Javier Fernán-dez Navas, por toda la dedicación y esfuerzo que han realizado para llevar a caboeste trabajo. Sus conocimientos sin duda han resultado muy valiosos durante larealización de este proyecto.

También me gustaría agradecer a toda mi familia, en especial mi madre Engra-cia García Arnedo y a mi padre Antonio Catalán Fernández, que con su pacienciahan aguantado todas las intempestades y me han motivado en este largo camino.

Por supuesto hay alguien que requiere mención especial, porque hace que juegecon ventaja en la vida, porque es un gran apoyo moral, muchas gracias tambiéna mi hermano, Jose Antonio Catalán García.

A todos, muchas gracias.

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Contents

1 Introduction 5

2 The general Kristensen-Mele approach 6

2.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Yang’s expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Risk-neutral probabilities . . . . . . . . . . . . . . . . . . . . . . . 82.4 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Pricing Results 9

3.1 The auxiliary model . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Models with stochastic volatility . . . . . . . . . . . . . . . . . . . 11

3.2.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . 113.2.2 Pricing results . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Models with jumps and stochastic volatility . . . . . . . . . . . . 153.3.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . 153.3.2 Pricing results . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Conclusions 22

A Appendix 23

A.1 The infinitesimal generator of the option price . . . . . . . . . . . 23A.2 Option Pricing Differential Equation . . . . . . . . . . . . . . . . 23A.3 Mispricing function . . . . . . . . . . . . . . . . . . . . . . . . . . 24A.4 The Feynman-Kac Theorem . . . . . . . . . . . . . . . . . . . . . 24A.5 Yang’s expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.6 Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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Abstract

This paper describes an approximation derivative pricing model recentlyintroduced in Kristensen and Mele (2011) (K-M, from now on) and illus-trates its applications to different derivatives and continuous-time models.This pricing model allows to obtain approximate asset prices for models thatdo not provide closed-form expressions in terms of the analytical formulasobtained for the price of these assets under a second (auxiliary) model.

Then, this approximation pricing model is based on the choice of an“auxiliary” pricing model for which a closed-form expression for prices isavailable. Given such auxiliary model, K-M derives an expression for thedifference between the unknown price of the model of interest (true model)and that (known) in the auxiliary model. Such expression takes the form ofa conditional expectation and can be developed in terms of a Taylor seriesexpansion.

After describing this model, we illustrate its implementation to someequity derivatives and models that reflect stochastic volatility and jumps.

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1 Introduction

During the last years, the use of continuous-time models have increased consid-erably and a large number of models have arisen. Continuous-time modeling hasprovided many benefits and it allows us to find elegant representations for theprice of a variety of contingent claims.

While many models have arisen, new approaches have also been developed andthey provided us new methods with applications to pricing, hedging, and span-ning of derivatives contracts. In some cases, the approaches produce closed-formsolutions to cope with prices. In some others cases where the closed-form doesnot exist, some other methodologies (as approximations or numerical approaches)have been developed to compute prices.

Although the benefits of the continuous-time modeling have been proved, anold issue remains:

How can we implement those methods that cannot be solved in closed-form?

Up to now, to deal with prices that are not given in a closed-form, severalalternatives have been proposed. We can mention Fourier transformations, treemethods, numerical solutions to partial differential equations or Monte Carlomethods, among others. While the implementation of some of these methods canbe cumbersome, a new approach has been recently proposed in Kristensen andMele (2011) (K-M, from now on).

These authors have developed a new approach to approximate prices in multi-factor models. This approach develops what could be interpreted as a closed-formprice formula for each multi-factor model. The key idea underlying this approachis to set as benchmark an auxiliary model for which a closed-form solution isknown. Once the auxiliary model is set, the goal of the K-M approach is to relatethe price under this auxiliary model to the price obtained from other model whichwill be considered as the target price.

This method will provide an expression for the difference between the modelof interest (which price is not known in closed-form) and the auxiliary one (whichis known in closed-form). Although such expression will take the form of a con-ditional expectation, K-M finds an expression that can be developed through aTaylor series expansion. Because this series expansion contains a infinite numberof terms, we will need to truncate the series into a finite number of terms.

This paper is organized as follows. Section 2 introduces the K-M approachin the most general case and discusses its similarities with respect to the Yang’sexpansion (see [7]) and the risk-neutral probabilities. We also discuss the appli-cation of this method to compute sensitivities (“Greeks”). Section 3 applies theK-M approach to different derivatives and models. First we will implement the

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method when the model of interest contains new information respect to the auxil-iary model. We also illustrate the K-M method and its properties under the mostextreme conditions. We will illustrate how these conditions affect the percentageerrors obtained through this approach and implement the K-M method usingdifferent number of terms in the series expansion. We also implement the modelunder a new and richer model than in the previous case, present an alternativeto the K-M approach and, because of the complexity of this model, we will showits limitations. Finally, Section 4 summarizes our main pricing results and therobustness of this method. Proofs and technical details are deferred to a finalAppendix.

2 The general Kristensen-Mele approach

2.1 Theoretical framework

To introduce the reader into our framework, our first benchmark is to set a truemodel, which produces our objective price. In this section we will consider ourtrue model as the most general model. Let x(t) be a d-dimensional multi-factormodel. Under the risk-neutral probability, we assume x(t), such that follows

dx(t) = µ(x(t), t)dt+ σ(x(t), t)dW (t), (2.1.1)

where W (t) is a standard Brownian motion under risk-neutral probability.First of all, we set w(x, t) as the option price ∀t ∈ [0, T ]. Besides, the existence

of the L̂ for any d-dimensional multifactor model drives us into a differentialequation, which is satisfied by the option price w(x, t) and it takes the form

Lw(x, t) + c(x, t) = R(x, t)w(x, t), (2.1.2)

where the option price is subject to the boundary condition at maturity timew(x, T ) = b(x), where b(x) is the payoff function of our derivative.

After the true model is presented, it is the moment to introduce the auxiliarymodel, which has an option price given by w0(x0, t). As we mentioned before,the option price for the auxiliary model also satisfies the equation 2.1.2, but withw0(x, t) replacing w(x, t) and with L̂0 replacing L̂ (in this case, the boundarycondition at maturity time is w0(x, T ) = b0(x).

Once the models are set, if we define the instantaneous price difference betweenthe two models as ∆w(x, t) = w(x, t) − w0(x, t) and then taking thee differencebetween the equation 2.1.2 and the corresponding to the auxiliary model, we find:

L∆w(x, t)− r∆w(x, t) + δ(x, t) = 0, (2.1.3)

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where the term δ(x, t) is known as the mispricing function and takes the form

δ(x, t) = (L− L0)w0(x, t). (2.1.4)

The above equation could be developed and the mispricing function turns into:

δ(x, t) =d

∑

i=1

∆µi(x, t)∂w0(x, t)

∂xi

+1

2

d∑

i,j=1

∆σ2ij(x, t)

∂2w0(x, t)

∂xi∂xj

, (2.1.5)

where ∆µ and ∆σ are the differences for the drift and diffusion terms betweenthe true and the auxiliary model, respectively.

Now, we can apply the Feynman-Kac representation to find a solution for theprice difference (see section A.4) and then the following identity holds:

∆w(x, t) = Ex,t

[

exp

(

−∫ T

t

R (x(s), s) ds

)

d (x(t))

]

+

+

∫ T

t

Ex,t

[

exp

(

−∫ s

t

R(xu, u)du

)

δ (x(s), s)

]

ds, (2.1.6)

where d(x) = b(x)− b0(x).The right-hand side of the above equation shows two terms of errors owing

to the use of an auxiliary model. The first term arises due to the use of differ-ent payoffs between true and auxiliary models (that difference is discounted tothe present value through the exponential function). The second term appearsbecause of differences related to the underlying factors of the models.

Taking advantage of the Appendix A of the reference [3], the above equationcan be written as a series expansion

wN(x, t) = w0(x, t) +∞∑

n=0

(T − t)n

n!dn(x, t) +

∞∑

n=0

(T − t)n+1

(n+ 1)!δn(x, t). (2.1.7)

In practice, this formula is truncated to a finite number of terms, yielding:

wN(x, t) = w0(x, t) +N∑

n=0

(T − t)n

n!dn(x, t) +

N∑

n=0

(T − t)n+1

(n+ 1)!δn(x, t), (2.1.8)

where δ0 = δ(x), d0 = d(x) and the δn function as well as the dn function satisfythe recursive equations:

dn(x, t) = Ldn−1(x, t)−R(x, t)dn−1(x, t), (2.1.9)

δn(x, t) = Lδn−1(x, t)−R(x, t)δn−1(x, t). (2.1.10)

Since w0 is known in her closed form, we are able to approximate (throughthe equation 2.1.8) the price of any option or also partial derivatives respect tothe variables of interest.

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2.2 Yang’s expansion

In this section, we will deal with the relationship with other expansion similarto Kristensen-Mele’s expansion (see Yang’s expansion through [7]). Under someconditions (see section A.5), the Yang’s expansion suggests that the differencebetween the price in the true market and in the auxiliary one satisfies

w(x, t) = w(0)(x, t) +

∫ T

t

E0x,t [(L− L0)w (x(u), u)] du. (2.2.1)

On the other hand, under simplifying assumptions done in Yang’s expansionto the Kristensen-Mele method (R(x, t) = c(x, t) ≡ 0 and d(x) = 0), then, theequation 2.1.6 drives into a similar but quite different price representation:

w(x, t) = w0(x, t) +

∫ T

t

E [(L− L0)w0 (x(u), u)] du. (2.2.2)

A quick look at both expressions might have some likenesses, but insteadlook over likenesses, we focus on the differences. Although both representationsseem to be close, the Yang’s expansion is based on the expectation under thebase-model’s probability, while K-M expansion is under the auxiliary model’sprobability. It is also quite important to note that Yang’s expansion is based ona unknown closed-form function, (L − L0)w, while that in K-M is based on awell known closed-form function, due to the use of an auxiliary model, δ = (L−L0)w0. For those reasons, the K-M expansion method allow us to approximatedirectly the integrated expectation, while Yang’s expansion needs the Feynman-Kac representations to be solved, which can be hard to implement.

2.3 Risk-neutral probabilities

Let p and p0 be the risk-neutral conditional densities underlying the true model(p) and the auxiliary model (p0) and considering that the payoff functions are thesame in both models (d(x) = 0). Then the two prices are

w(x, t) =

∫

Rd

b(y)p(y, T |x, t)dy,

w0(x, t) =

∫

Rd

b(y)p0(y, T |x, t)dy.(2.3.1)

Taking the difference of these two different prices, we find

w(x, t) = w0(x, t) +

∫

Rd

b(y)∆p(y, T |x, t)dy. (2.3.2)

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Now, it is easy to see such relation with the K-M method. First, let assumeR = 0, c = 0 and d(X) = 0. Those assumptions set the first term in the rightside of equation 2.1.6 equal to zero, the exponential of the second term equalto one, and hence the Kristensen-Mele’s method relates with that of probabilitydensities as follows:

∫

Rd

b(y)∆p(y, t|x, t)dy =

∫ T

t

Ex,t [δ(x(s), s)] , (2.3.3)

where δ has the from of eq. 2.1.5. The right-hand of equation 2.3.3 can besolved through a power series expansion as shown in equation 2.1.8, while therisk-neutral method could be solved through Riemann integral, which can becumbersome to implement (particularly when dimensionality of our model grows,the implementation becomes harder).

2.4 Greeks

Once equation 2.1.8 is given explicitly, the partial derivatives with respect to thevariables of interest can be easily derived. Hence, the k-th order derivative ofw(x, t) is given by:

∂wN(x, t)

∂xk=

∂w0(x, t)

∂xk+

N∑

n=0

(T − t)n

n!d(k)n (x, t)+

N∑

n=0

(T − t)n+1

(n+ 1)!δ(k)n (x, t), (2.4.1)

where

d(k)n (x, t) =∂kdn(x, t)

∂xk,

δ(k)n (x, t) =∂kδn(x, t)

∂xk.

These terms have been developed until second order of derivative in [3].

3 Pricing Results

The main goal of the next sections will be to validate the K-M method throughpricing results. We will analyze the method and the sensitivity under differentcontexts and we will depict the error for each context. It is also quite interestinghow many terms we need in order to obtain good results.

First of all, we set our scenario for the auxiliary and the true models. In allour cases, the auxiliary model will be the Black-Scholes, while the true model willbe selected with different purposes, but all of them adding new more information

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on a sequential basis. First we will consider a richer model than the Black-Scholes model, which adds new information in terms of stochastic volatility. Theselected model has been the CEV model (analyzed in [4]) which has a similarprocess than the Black-Scholes model, but including a stochastic process for thevolatility. This model has already been studied by K-M. Our objective will beto check our methodology and study new extreme scenarios that have not beenstudied by K-M, for example, when the option price is very close to zero.

After we test the approximation from Black-Scholes to CEV, a new richermodel will be presented. We have considered the Merton model with stochas-tic volatility as our last experiment, which contains information about JumpDiffusion and therefore add new more information than Black-Scholes and evenremaining the stochastic volatility information that was introduced by the CEVmodel. Although this method has been introduced in a theoretical framework byK-M, they have not studied this model empirically. We will study (empirically)this case under the theoretical framework proposed by K-M and under a newalternative that we will develop.

Once the structure has been explained, in the next section, our benchmarkwill be the auxiliary model and those properties that are related with our purpose.After the auxiliary model’s presentation, we will test the K-M approach throughthe models we have mentioned above.

3.1 The auxiliary model

In all our futures scenarios, we will develop the K-M approach through the sameauxiliary model, the Black-Scholes model (see [1]). We have chosen this one as theauxiliary model, because the Black-Scholes model provides a closed-form pricingformulas for a large number of derivatives and therefore this start point could beextended to any other derivatives and models.

The Black-Scholes model is a one-dimensional factor model where the statevariable is the stock price and is the solution to

dS(t)

S(t)= rdt+ σ0dW (t), (3.1.1)

where W (t) is a standard Brownian motion under the risk neutral probability, rand σ0 are the short-term rate and the volatility respectively, and both taken tobe constant.

The infinitesimal operator related to this model is given by

L̂0 =∂

∂t+ rS(t)

∂

∂S+

1

2σ20S(t)

2 ∂2

∂S2. (3.1.2)

In addition to that, in the whole paper, we will refer to w0(x, t) as the pricegiven by the auxiliary price, which should be known in closed form.

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3.2 Models with stochastic volatility

3.2.1 Theoretical framework

In this subsection we will consider the CEV model (see [2]) as the true model andBlack-Scholes remains as the auxiliary one. The CEV model is a 2-dimensionalmulti-factor model, so that x(t) is in this case, the vector [S(t) v(t) ]. The explicitformula that follows the CEV model satisfies

dS(t)

S(t)= rdt+

√

v(t)dW (t), (3.2.1)

where W (t) is a standard Brownian motion under risk-neutral probability, r isthe short-term rate (taken to be constant) and the variance follows the belowstochastic process

dv(t) = κ(α− v(t))dt+ ω|v(t)|ξdWv(t), (3.2.2)

where Wv(t) is a standard Brownian motion correlated with W (t) with instanta-neous correlation ρ, ξ < 0 is the CEV parameter and (κ, α, ω) are three additionalconstants.

In terms of the infinitesimal generator, the CEV model has an operator whichsatisfies

L̂ =∂

∂t+ S(t) · r ∂

∂S+ κ(α− v(t))

∂

∂v+

1

2S(t)2v(t)

∂2

∂S2+

+1

2ω2v(t)2ξ

∂2

∂v2+ ρωS(t)v(t)ξ+1/2 ∂2

∂S∂v. (3.2.3)

The above expression redefines the “mispricing function”, δ, (see eq. 2.1.4)which is given by

δ(x, v, t; σ0) =1

2(v − σ2

0)S2∂

2w0

∂S2, (3.2.4)

since w0(S, v, t; σ0) is known in closed form, we can compute directly the mis-pricing function. As we mentioned before, the recursive function remains thesame

δn(x, t) = L̂δn−1(x, t)−R(x, t)δn−1(x, t), (3.2.5)

and the initial condition is δ0 = δ(x, t).

3.2.2 Pricing results

Once the code has been developed, the validation of the Kristense-Mele approachwe will be analysed through different ways.

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S CEV MCKristensen-Mele Kristensen-Mele

N=0 N=4950 57.5661 58.0455 57.7312960 62.1077 62.5759 62.2049970 66.8486 67.3055 66.8835980 71.7846 72.2332 71.7666990 76.9181 77.3568 76.85321000 82.2477 82.6742 82.14181010 87.7666 88.1826 87.63001020 93.4722 93.879 93.31531030 99.3647 99.7601 99.19431040 105.443 105.822 105.2631050 111.702 112.061 111.518

Table 1: European call option prices for the CEV model computed through MonteCarlo simulations and through the K-M method. Variations on stock price. Thevolatility parameter is set equal to v(t) = 0, 5172.

Tables 1 and 2 compare European call option prices for the CEV model com-puted through Monte Carlo simulations and through the K-M method. Europeancall options have a strike of K = 1000, time to expiration equals to one month(T − t = 1/12), and the parameters of the equations 3.2.1 and 3.2.2 are set equalto κ = 0.1465, α = 0, 5172, ω = 0.5786, ξ = 0.6, ρ = −0.0243, and r = 0. Panel 1produces option prices when the initial value of the instantaneous variance equalsto v(t) = α. Panel 2 produces option prices when the initial value of the underly-ing stock price equals S(t) = 1000. Both panels provide the prices of our methodfor different number of leading terms of eq. 2.1.8.

Tables 1 and 2 also show how the prices accurate as we reach high terms(N = 4) of our expansion (see eq. 2.1.8). As we can see, the price given bythe Kristensen-Mele’s method tends to converge to the true price1 (CEV). Thesetables also show how the K-M approach captures the variations through differentvariables of interest, in our case, the volatility v(t) and the stock price S(t) (inboth cases show a high level of confidence).

Tables 1 and 2 also show how the prices accurate as we reach high terms(N = 4) of our expansion (see eq. 2.1.8). As we can see, the price given bythe K-M method tends to converge to the true price2 (CEV). These Tables also

1The option price computed through Monte Carlo simulation have been designed with 50000number of paths.

2The option price computed through Monte Carlo integration have been designed so that we

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show how the K-M approach captures the variations through different variablesof interest, in our case, the volatility v(t) and the stock price S(t) (in both casesshow a high level of confidence).

On the other hand, figures 1a, 1b and 1c show the joint variation of thepercentage error of the K-M method respect to the stock price S(t) and respectto the variance v(t). In those figures, the behavior of the percentage pricing erroris quite different for option with in-the-money stock price from those with out-of-the-money stock price. First, attending to in-the-money stock prices, we concludethat the K-M’s method produces good results in a large range of volatility, beingthe percentage pricing errors around 0.5% for N = 0, a percentage error of 0.4%for N = 1 and lower than 0.1 for N = 4. In contrast to in-the-money options,the options with out of the money stock price produces percentage errors around1% when N = 0, a percentage error of 0.8% when N = 1 and 0.5% when N = 4.A special case, which has not studied by [3], is when we attend to far-out-of-the-money options (stock prices between 850 and 950). The further-out-of-the-moneyoptions produces higher percentage errors, specially when the volatility is closeto zero. It happens because the option prices are very close to zero and due tothe definition of the percentage error.

From these results, we might confirm that the K-M approach produces a goodclosed pricing formula and we have also proved that the L̂ operator add newinformation about the true model as the operator iterates upon the mispricingfunction so that only a very few terms of eq. 2.1.8 are needed to provide quiteaccurate approximations to the unknown price.

keep less than 0.5% discrepancy between the price computed through Monte Carlo integrationand the prices obtained through Fourier transforms.

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850900

9501000

10501100

1150

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

underlyingvolatility

%D

iff

(a)

850900

9501000

10501100

1150

0.20.4

0.60.8

1

0

0.2

0.4

0.6

0.8

1

1.2


%D

iff

(b)

850900

9501000

10501100

1150

0.2

0.4

0.6

0.8

1

−1

0

1

2

3

4

5

6


%D

iff

(c)

Figure 1: Joint variation of the percentage error of the Kristensen-Mele’s methodwith respect to the stock price S(t) and to the variance v(t) when N = 0 (figure1a), N = 1 (figure 1b) and N = 4 (figure 1c).

14

v CEV MCKristensen-Mele Kristensen-Mele

N=0 N=40.1 36.006 36.4058 35.55470.2 51.0693 51.4676 50.76660.3 62.6054 63.0128 62.3890.4 72.3192 72.7357 72.16180.5 80.8676 81.2927 80.7550.6 88.5887 89.0208 88.5110.7 95.6825 96.1201 95.63270.8 102.279 102.721 102.2520.9 108.470 108.914 108.4621.0 114.320 114.766 114.3271.1 119.885 120.326 119.899

Table 2: European Call Option prices for the CEV model computed throughMonte Carlo and through the K-M method. Variations in volatility. The stockprice is set to S(t) = 1000.

3.3 Models with jumps and stochastic volatility

Finally, a richer model will be studied. The K-M approach will be tested with amodel which includes jump diffusion as well as stochastic volatility.

3.3.1 Theoretical framework

The jump-diffusion model with stochastic volatility that will be analyzed is theMerton jump-diffusion (76) model (see [5] and [6]) and the stochastic volatilityfollows a CEV process.

dS(t)

S(t)= (r − λj̄)dt+

√

v(t)dW (t) + jdP (t), (3.3.1)

where W (t) is a Brownian motion, dP (t) is a Poisson process with a boundedintensity parameter of λ, j is a random bariable with probability of measure on[−1,∞), density p and expectation j̄. The stochastic process that our volatilityfollows is a CEV model, that is, v(t) evolves according to

d(v) = κ(α− v(t))dt+ ω|v(t)|ξdWv(t), (3.3.2)

where Wv(t) is a Brownian motion correlated with W (t) with correlation ρ, ξ > 0is the CEV parameter and (κ, α, ω) are three additional constants.

This model has been developed through two similar ways, so that we couldtest the robustness of the K-M approach and the limits of this method.

15

• First method

First, we set the infinitesimal generator related to the Merton-CEV modelas K-M suggests in [3]. In this case, the L̂ operator is given by the followingexpression

L̂Jf(x, v, t) = L̂f(x, v, t)+λ

∫

∞

−1

[f(x(1+ j), v, t)− f(x, v, t)]p(dj), (3.3.3)

where the L̂ operator takes the form of the equation 3.2.3.

Since eq. 3.3.3 split the expression of the L̂J operator into the L̂ operator,eq. 3.2.3 (which contains stochastic volatility information) and a secondterm (which contains jump-diffusion information through the integral), ourmispricing function function is simply the eq. 3.2.4, but adding the righthand term of eq. 3.3.3. The mispricing function is now

δ(x, v, t; σ0) =1

2(v−σ2

0)S2∂

2w0

∂S2+λ

∫

∞

−1

[w0(x(1+j), v, t)−w0(x, v, t)]p(dj),

(3.3.4)where δn follows the recursive function

δn(x, t) = L̂Jδn−1(x, t)−R(x, t)δn−1(x, t). (3.3.5)

• Second method

As a second method, we present an alternative to the previous methodwhich is related to the L̂ operator through the Itô’s lemma, and takes theform of:

L̂J = (r − λj̄)S(t)∂

∂S+ κ(α− v(t))

∂

∂v(t)+

∂

∂t+

+1

2S(t)2(v(t) + j2λ)

∂2

∂S2+

1

2ω2|v(t)|2ξ ∂2

∂v2+

+ρS(t)v(t)1/2ω|v(t)|ξ ∂2

∂S∂v, (3.3.6)

where j2 is j2 = E[j2] = var[j] + E[j]2 = σ2j + j̄2.

Once the explicit expression for the L operator is known and consideringBlack-Scholes as the auxiliary model as we did in previous section, we areable to compute the recursive function

δn(x, t) = L̂Jδn−1(x, t)−R(x, t)δn−1(x, t), (3.3.7)

16

where L̂J behaves liking in equation 3.3.6 and the mispricing function is thesolution to

δ0 = (L̂− L̂0)w0 = −λj̄S(t)∂w0

∂S+ κ(α− v(t))

∂w0

∂v(t)+

+1

2S(t)2(v(t) + (σ2

j + j̄2)λ− σ20)∂2w0

∂S2+

1

2ω2|v(t)|2ξ ∂

2w0

∂v2+

+ρS(t)v(t)1/2ω|v(t)|ξ ∂2w0

∂S∂v,

where L̂0 takes the form of 3.1.2. Because of the Black-Scholes model doesnot depend on the v parameter (it depends on the σ0), all the partial deriva-tives with respect to v are equal to zero, that is ∂w0/∂v = ∂2w0/∂v

2 =∂2w0/∂Sv = 0. Under these assumptions, the mispricing function formulafor the Merton-CEV model is given by

δ0 = −λj̄S(t)∂w0

∂S+

1

2S(t)2

(

v(t) + (σ2j + j̄2)λ− σ2

0

) ∂2w0

∂S2. (3.3.8)

Once our objectives have been set, we will compare our method with that ofthe K-M approach.

3.3.2 Pricing results

As in Section 3.2.2, we will analyze the K-M approach through different methods.First we will depict the percentage approximation error as a function of the stockprice and the volatility in 3D plots. Later on, a table will provide explicit pricesgiven by our method for many different leading terms. This Table comparesthe performance of our approximations with those of Merton with stochasticvolatility.

In figures 2 and 3, options are European call options. They have a strikeprice of K = 1000, time to maturity equals one month (T − t = 1/12), andthe parameters of equations 3.3.1 and 3.3.2 are set equal to κ = 1, ω = 0.5786,ξ = 0.6, ρ = −0.5, λ = 1, z̄ = 0 and σz = 0.4. Where z is distributed as aNormal distribution and therefore j = (exp(z)−1) is distributed with probabilityof measure [−1,∞). We have also set α equals to the instantaneous variance v(t)and the volatility of the Black-Scholes model equals to σ0 =

√v.

As we can see, Figures 2 and 3 depict the approximation errors resulting fromour method for the two alternatives we presented in the previous section. Onceagain, both methods provides better results as we add new more terms. In thiscase, the results could only be tested until first leading term, and hence, wecan not confirm a strong evidence of convergence to the Merton with stochastic

17

950

1000

1050

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0

2

4

6

8

10

12

14

16


%Di

ff

(a)

950

1000

1050

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

−5

−4.5

−4

−3.5

−3

−2.5

−2


%Di

ff

(b)

Figure 2: Joint variation of the percentage error of our method respect to thestock price S(t) and respect to the variance v(t) when N = 0 (figure 1a), N = 1(figure 1b).

18

950

1000

1050

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0

2

4

6

8

10

12

14

16


%Di

ff

(a)

950

1000

1050

0.2

0.4

0.6

0.8

1

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5


%D

iff

(b)

Figure 3: Joint variation of the percentage error of our method respect to thestock price S(t) and respect to the variance v(t) when N = 0 (figure 1a), N = 1(figure 1b).

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SMerton-CEV Our alternative Kristensen-Mele

MC N=1 N=1950 59,5270 57,0572 56,5877960 64,1573 61,4851 61,0106970 68,9898 66,1588 65,6809980 74,0289 71,078 70,5982990 79,2761 76,2412 75,76091000 84,7222 81,6453 81,1661010 90,3661 87,2864 86,80961020 96,2039 93,1595 92,68671030 102,235 99,2588 98,79121040 108,456 105,577 105,1171050 114,854 112,108 111,655

Table 3: European call option prices for the Merton-CEV model computedthrough Monte Carlo and through the K-M method. Variations in stock price.The variance parameter is set to v(t) = 0.58.

volatility prices. Nevertheless, the percentage errors are substantially cuttingdown when only a new one leading term is added.

With the purpose of explaining why we only could develop our methods untilfirst order, we might take a look to the number of partial derivatives our L̂J

operator contains in eq. 3.3.3 as well as in eq. 3.3.6. These partial derivativeshave been computed through central finite differences until fourth order becausethe second order approximation bounded us into errors (see A.6). Attendingto the central finite differences method until fourth order, each single partialderivatives iterates four times upon the recursive function, each second partialderivative iterates five times, the cross partial derivatives iterates eight times andthe independent term iterates once. All in all, when the recursive function acts,it makes our method cumbersome, and in some cases, when N becomes higher,we are limited by the computer’s memory. For instance, if we want to developthe eq. 2.1.8 until first order, we calculate 27 evaluations over the mispricingfunction, for N = 2 we need 272 = 729 and in general, for N = i we need 27i

evaluations. Under this assumptions, with only a very few leading terms in oureq. 2.1.8, we reach millions of evaluations.

In addition to that, eq. 3.3.6 and the mispricing function (see eq. 3.3.8)contain integrals, which needs to be calculated by numerical method (in our casecomputed by the rectangle rule). Although the integral term seems to be a simpleintegral, we might attend when L̂J acts over the recursive function in eq. 3.3.7

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vMerton-CEV Our Alternative Kristensen-Mele

MC N=1 N=10,3 72,0123 68,9270 68,33780,42 77,7557 74,7045 74,16950,54 83,0434 79,9767 79,48490,66 87,9711 84,8695 84,41320,78 92,6028 89,4613 89,03490,90 96,9868 93,8053 93,40451,02 101,159 97,9398 97,56131,10 103,838 100,594 100,229

Table 4: European call option prices for the Merton-CEV model computedthrough Monte Carlo simulations and through the Kristensen-Mele’s method.Variations on volatility. Parameter value of stock price is S(t) = 1000.

as well as when it does over the mispricing function (see eq. 3.3.8). When theintegral needs to be calculated, we have to choose between high precision, whichmeans a high number of iterations, or either a lower number of iterations, whichmight lead to numerical errors.

In both scenarios we built, we have tried to test the K-M approach, andthe results are quite interesting. Although both methods seems to reduce thepercentage price errors as we add new more terms, our method performs betterthan K-M suggestion, unless when first order of approximation are developed.

On the other hand, Tables 3 and 4 compares the performance of the K-Mapproach. The option prices for Merton-CEV model have been computed throughMonte Carlo simulation. Table 3 provides option price variations respect to thestock price, while table 4 does respect to the variance variable.

In both Tables, our alternative through eq. 3.3.6, provides better resultsthan those of K-M and the percentage difference error have been reduced whenoptions are in-the-money as well as far-out-of-the-money. As we saw with theCEV model, the percentage error tends to increase as volatility tends to lowervalues, and also as prices go far-out-of-the-money (due to the low value of price).

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4 Conclusions

This paper has analyzed the K-M approach and illustrated its implementationfor different derivatives and models.

In our first case, where the CEV model was chosen as true model, we havestudied how new terms in the series expansion of eq. 2.1.8 add new more in-formation about the true model and they capture the stochastic behavior of ourobjective model. We have also confirmed that we can obtain quite accurate ap-proximations to the unknown price, with only a very few additional correctiveterms to eq. 2.1.8.

For this scenario, with four leading terms, the K-M approach includes enoughadditional information about the stochastic volatility and also seems to be thebest choice in terms of computational time.

We have also shown that the percentage errors increase when options are veryfar-out-of-the-money. This behavior is due to the definition of the percentageerror and the fact that prices are very close to zero.

In our second scenario, with the Jump Diffusion model and stochastic volatil-ity as true model, the results are quite interesting, despite we have been forcedto face off some numerical issues because the numerical computation becomesharder.

One of them was the error propagation through the iterations when the re-cursive function is applied. As central finite differences until second order doesnot produce accurate prices and the errors propagate through the iterations, wehave extend our approximation to derivatives through central finite differencesuntil fourth order. They have minimized the propagation of the errors and pricesseem to converge to the true price.

Because of computational limitations, the computation of the integral in eq.3.3.3 does not allow us to calculate more terms in the series expansion than thetwo first leading terms. Although we only could develop our results until thetwo first leading term, some interesting conclusions could be obtained. As weadd these two leading terms (δ0 and δ1), prices seem to converge to the price ofinterest.

We have introduced an alternative to the approximation suggested by the K-M approach. This alternative is also computationally cumbersome but we do notdeal with the propagation of errors through the iterations of the integral. Ouralternative decreases the computational time with respect to the K-M suggestion.Since we can avoid the errors related to the integral, the results might be moreaccurate.

In this section, we were computationally limited and the choice of anothermodel as the auxiliary model could make it lighter. It might be for instance, theMerton model, which only needs to add information about stochastic volatility.

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A Appendix

A.1 The infinitesimal generator of the option price

Let our model be associated with equations 3.2.1 and 3.2.2. Considering the Itô’slemma and let w(t) = w(S, v, t) be the option price function, then the infinitesimalprocess of the option price follows

dw(t) =[

S(t) · r∂w∂S

+ κ(α− v(t)) +1

2S(t)2v(t)

∂2w

∂S2

+1

2ω2v(t)2ξ

∂2w

∂v2+ ρωS(t)v(t)ξ+1/2 ∂2w

∂S∂v+

∂w

∂t

]

dt+

+S(t)√

v(t)∂w

∂SdW (t) + ω|v(t)|ξ ∂w

∂vdWv(t).

In operator’s notation holds:

dw(t) = L̂wdt+ M̂wdW (t) + N̂wdWv(t), (A.1.1)

where the L̂ operator is considered as the Heston’s infinitesimal generator.Any other model could be proved in the same way.

A.2 Option Pricing Differential Equation

Let π be our portfolio and assuming it is a hedge portfolio composed by optionsof interest and the underlying

dπ = k1dw + k2dS. (A.2.1)

If we suppose Heston as our model and therefore

dS = rSdt+√

v(t)SdW (t).

As shown in the previous section

dw(t) = L̂wdt+ M̂wdW (t) + N̂wdWv(t).

Replacing dS and dw(t) from the equation A.2.1, and applying discretizationmethods,

∆π = [k1Lw + k2rS] ∆t+[

k1Mw +K2

√

v(t)S]

∆W (t) + k1Nw∆Wv(t),

where we set k1 = −1. Assuming that risk can be protected against with a hedge,

k2 =Mw

√

v(t)S≡ ws. (A.2.2)

23

Assuming our portfolio is a hedge portfolio, then ∆π = rπ∆t, and also as-suming that the diffusion term of the variance can not be hedge, we get:

r [−w + wsS] = −Lw + wsrS,

Lw(S, v, t) = rw(S, v, t). (A.2.3)

The differential equation has been proved for the CEV model. Any othermodel follows the same process.

A.3 Mispricing function

As we obtained the differential equation A.2.3, for any other models

L0w0(S, v, t)− rw0(S, v, t) = 0.

Taking the difference between the above equation and the equation A.2.3, itis trivial to find the explicit form of the mispricing function

Lw − rw − (L0w0 − rw0) = L∆w − r∆w + (L− L0)w0 = 0,

and therefore

L∆w(S, v, t; σ0)− r∆w(S, v, t; σ0) + δ(S, v, t; σ0) = 0, (A.3.1)

where ∆w = w(S, v, t)− w0(S, t; σ0) and δ = (L− L0)w0.

A.4 The Feynman-Kac Theorem

Stochastic differential equations are closely linked to partial differential equations.The Feynman-Kac theorem is key to Financial modeling.

The Feynman-Kac Representation 1 Let f ∈ C1,2 be a solution to the de-terministic partial differential equation:

∂

∂tf(t, x) + µ(t, x)

∂

∂xf(t, x) +

1

2σ2(t, x)

∂2

∂x2f(t, x)− v(x)f(t, x) + g(t, x) = 0,

(A.4.1)for all (t, x) ∈ [0, T ] subject to the terminal condition f(T, x) = h(x). Let thesolution to the stochastic differential equation:

dXt = µ(s,Xs)ds+ σ(s,Xs)dWs,

where s ∈ [t, T ] and Xt = x. Then,

f(T, x) = E

[

exp

(

−∫ T

t

v(s,Xs)ds

)

h(XT )|Xt = x

]

+

+

∫ T

t

E

[

exp

(

−∫ s

t

v(u,Xu)du

)

g(s,Xs)|Xt = x

]

ds

(A.4.2)

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A.5 Yang’s expansion

For the purpose of simplifying these section, we will assume R(x, t) = c(x, t) ≡ 0and that the payoff function are the same for both models (implying d(x) = 0).Yang’s expansion starts with a base-model, which satisfies L0w

(0) = 0. Using theequation A.2.3, the unknown price can be written as the solution to

0 = Lw(x, t) ≡ L0w(x, t) + (L− L0)w(x, t). (A.5.1)

The fact that L0w(0) = 0 allows us to rewrite the above equation to a new one

L0∆w(x, t) + (L− L0)w(x, t) = 0, (A.5.2)

and its solutions satisfies:

w(x, t) = w(0)(x, t) +

∫ T

t

E0x,t [(L− L0)w(x(u), u)] du. (A.5.3)

A.6 Finite Difference

The approximation of derivatives by finite differences plays a central role in finitedifference methods for the numerical solution of differential equations. We willdevelop the central finite differences for first and second derivatives until fourthorder.

1. Finite differences for first derivatives.

Central finite difference scheme second order:

∂f(x)

∂x=

f(x(1 + h))− f(x(1− h))

2xh+ ǫ(2).

Central finite difference scheme fourth order:

∂f(x)

∂x=

−f(x(1 + 2h)) + 8f(x(1 + h))− 8f(x(1− h)) + f(x(1− 2h))

12xh+ǫ(2).

2. Finite differences for second derivatives.

Central finite difference scheme second order:

∂2f(x)

∂x2=

f(x(1 + h))− 2f(x) + f(x(1− h))

x2h2+ ǫ(4).

Central finite difference scheme fourth order:

∂f(x)

∂x=

−f(x(1 + 2h)) + 16f(x(1 + h))− 30f(x)

12x2h2+

+16f(x(1− h))− f(x(1− 2h))

12x2h2+ ǫ(4).

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References

[1] F. Black and M. Scholes. The pricing of options and corporate liabilities.Journal of Political Economy 81, 637-659, 1973.

[2] S. Heston. A closed-form solution for options with stochastic volatility withapplications to bond and currency options. Review of Financial Studies 6,327-344, 1993.

[3] D. Kristensen and A. Mele. Adding and subtracting Black-Scholes: A new ap-proach to approximating derivative prices in continuous-time models. Journalof Financial Economics 102 (2011) 390-415, 30 June 2011.

[4] A.L Lewis. Option Valuation Under Stochastic Volatility. Finance Press,Newport Beach, CA, 2000.

[5] Robert C. Merton. Option pricing when underlying stock returns are dis-continuous. Massachusetts Institute of Technology. Cambridge, Mass.02139,U.S.A. Journal of Financial Economics 3 (1976) 125-144, 1975.

[6] Javier F. Navas. Calculation of Volatility in a Jump-Diffusion Model. TheJournal of Derivatives 11, 2, 66-72, 2003.

[7] J. Yang. Stochastic Volatility Models: Option Price Approximations. Asymp-totics and Maximum Likelihood Estimation. PhD. Disertation, University ofIllinois at Urbana-Champaign. 2006.

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